3 Answers
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It's worth mentioning that even though at first the major cycling events were strictly individual, soon teams started emerging in a "natural", free market way. The stronger riders, who had a real chance of winning, began to pay off the weaker competitors to serve as domestiques.

This was at first considered "underhanded", like Marius Balaban mentions, and the cycling federations fought it at first. What happened later can be pictured as a win for reason, common sense and free market, or as a travesty of noble sports ideals, depending on how you are inclined to view the world...

In the first editions of Tour de France individuals and teams were allowed to participate. Because of some events and underhand tactics of sponsored teams only national teams were allowed to participate between 1930 and 1961 (not sure about the period).
In 1962 trade teams returned.

In this interesting video about the history of the Tour de France, at some point the narrator explicitly mentions the period (during the 70s if I remember well) when great cycling events beginned to change from a one-to-one fight amongst a few super-strong riders (typical example: Merckx) to a teamwork fight amongst well-orchestrated, hierarchical teams (typical example: Armstrong).

I think (and the video also mentions it) that this is due, among other things, to an increased understanding - and acceptance - of the importance of aerodynamics (LeMond), and also to advances in scientific training, which leveled the differences between well-prepared riders, I think.

Also, the role of the team in one or another era can be inferred from the very title of another excelent video, "Stars and Watercarriers".